Background

This document has nls (non-linear least squares) regression fits using the log-normal functional form to USFS FIA (United States Forest Service Forest Inventory & Analysis) biomass growth vs. stand age relationships. This functional form is commonly used in growth analyses, and permits a flexible shape to fit to data with an intermediate maximum (i.e., “hump” shaped) curve. As in our models of biomass growth vs. biomass, we use the mass balance biomass growth method for the plot biomass growth (\(G\)) calculation (briefly, plot biomass growth is a function of the change in plot biomass plus any losses due to mortality or harvest over time: \(G_{MB} = (\Delta B + M_t + C_t) / REMPER\), where \(\Delta B\) is change in plot biomass over a census interval ( \(\Delta B = B_{t + \Delta g} - B_t\) ), and \(M_t\) and \(C_t\) is the biomass of trees that died or were harvested, respectively, between two plot measurements. note: \(REMPER\) is time between two plot measurement intervals (FIA re-measurement period). For additional details see supplementary methods. Models are fitted separately by US ecoprovince.

Hypothetically, the entire functional form of the following non-linear model is considered: \(G = (1 + (yr-1990) \cdot tau/100) \times (1 - \alpha \cdot B_l) \times \left(a + b \cdot \exp{ - \left[ \frac{ \log \left( StdAge_{t1} /c \right)} {d} \right]} ^2 \right)\), where \(G\) is the plot level biomass growth calculated as the sum of tree biomass growth increments, \(B_l\) is the calculated proportion of biomass loss over the census interval, \(StdAge_{t1}\) is the FIA-estimated stand age at the first of two FIA plot tree censuses, and \(yr\) is the measurement year (all FIA data). Free parameters are \(\alpha\): the growth compensation of lost plot biomass, \(\tau\): the productivity trend, \(a\): the y-intercept of the curve, \(a +b\): the peak value of \(G\), \(c\): the \(StdAge_{t1}\) value at peak \(G\), and \(d\): the curve shape parameter.

Data have increasing variance in \(G\) with increasing \(StdAge_{t1}\), Thus, weighted nls is the best approach. We explore a few weighting options and found that proportional weighting can be achieved by weighting observations by \(\frac {1} {StdAge_{t1}^2}\) in equal-sample sized plot biomass bins (n=20 where applicable, else n=10) for each ecoprovince. These bins are also used to visualize data means in relation to nls model fit.

Model selection is done to determine the best fitting models, considering the inclusion of \(\alpha\): the biomass compensation effect due to lost biomass (natural mortality or harvest). Thus, the following two models are considered:

model 1: simple (tau) model \(G = (1 + (yr-1990) \cdot tau/100) \times \left(a + b \cdot \exp{ - \left[ \frac{ \log \left( StdAge_{t1} /c \right)} {d} \right]} ^2 \right)\)

model 3: model \(G = (1 + (yr-1990) \cdot tau/100) \times (1 - \alpha \cdot B_l) \times \left(a + b \cdot \exp{ - \left[ \frac{ \log \left( StdAge_{t1} /c \right)} {d} \right]} ^2 \right)\)

NOTE:

This document contains all \(G\) observations that meet our plot based filtering criteria:

  1. exclude FIA plots in plantation forests
  2. exclude FIA plots with multiple plot conditions (COND_PROG_UNADJ > 0.95)
  3. exclude FIA plots non-productive stands (i.e., those with less than 20 ft^3/acre/year timber producing capability; SITECLCD of 7)
  4. exclude FIA plots in non-stocked stands (i.e., those with STDSZCD of 5)
  5. exclude FIA plots in non-accessible areas (i.e., private lands etc., COND_STATUS_CD not equal to 1)
  6. exclude FIA plot visits that are not part of the annual inventories (which also includes FIA plot visits for Phase 3 ozone measurements)

Additionally, in an effort to clean up the data set, we have removed outlier observations, using a quantile threshold approach. We also calculated plot \(G\) using as biomass balance method (see supplementary methods), and the difference between the two methods. Accordingly, we define \(diff_G\) as the difference between tree incremental \(G\) and biomass balance \(G\). We excluded observations which meet the following criteria using a 0.5% quantile (\(QT\)):

  • case A: where the \(QT\) difference in tree incremental \(G\) is > biomass balance plot G (i.e., > 99.5% \(diff_G\) positive outliers)

  • case B: where the \(QT\) difference in tree incremental \(G\) is < mass balance plot G (i.e., < 0.5% \(diff_G\) negative outliers)

  • case C: where the \(QT\) difference in tree incremental \(G\) is > 0 (i.e., > 99.5% positive outliers)

  • case D: where the \(QT\) difference in tree incremental \(G\) is > 0 (i.e., < 0.5% negative outliers)

These data set cleaning criteria resulted in the exclusion of 1760 observations.

Below the model fitting procedure is implemented by ecoprovince:

211 - Northeastern Mixed Forest

model selection 1

## Analysis of Variance Table
## 
## Model 1: G_MassBal_MgHaYr ~ (1 + (MEASTIME_avg - 1990) * tau/100) * (a + b * exp(-((log(STDAGE_t1/c))/d)^2))
## Model 2: G_MassBal_MgHaYr ~ (1 + (MEASTIME_avg - 1990) * tau/100) * (1 - alpha * B_L_prop) * (a + b * exp(-((log(STDAGE_t1/c))/d)^2))
##   Res.Df Res.Sum Sq Df Sum Sq F value    Pr(>F)    
## 1   6838     2148.3                                
## 2   6837     2052.6  1 95.655  318.61 < 2.2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##   model      AIC
## 1     1 27051.67
## 2     2 26741.98
## 
## Formula: G_MassBal_MgHaYr ~ (1 + (MEASTIME_avg - 1990) * tau/100) * (1 - 
##     alpha * B_L_prop) * (a + b * exp(-((log(STDAGE_t1/c))/d)^2))
## 
## Parameters:
##       Estimate Std. Error t value Pr(>|t|)    
## tau    0.31226    0.17582   1.776 0.075784 .  
## alpha  0.63852    0.03351  19.054  < 2e-16 ***
## a      0.00000    1.60068   0.000 1.000000    
## b      3.43281    1.59386   2.154 0.031294 *  
## c     34.47438    1.75397  19.655  < 2e-16 ***
## d      2.54310    0.73279   3.470 0.000523 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.5479 on 6837 degrees of freedom
## 
## Algorithm "port", convergence message: relative convergence (4)
##   (41 observations deleted due to missingness)

summary

  • simple tau model: fits
  • alpha model: fits

predict and plot

## Warning: Using `size` aesthetic for lines was deprecated in ggplot2 3.4.0.
## ℹ Please use `linewidth` instead.
## Warning: Removed 23 rows containing missing values (`geom_point()`).

plotting 2

212 - Laurentian Mixed Forest

model selection 1

## Analysis of Variance Table
## 
## Model 1: G_MassBal_MgHaYr ~ (1 + (MEASTIME_avg - 1990) * tau/100) * (a + b * exp(-((log(STDAGE_t1/c))/d)^2))
## Model 2: G_MassBal_MgHaYr ~ (1 + (MEASTIME_avg - 1990) * tau/100) * (1 - alpha * B_L_prop) * (a + b * exp(-((log(STDAGE_t1/c))/d)^2))
##   Res.Df Res.Sum Sq Df Sum Sq F value    Pr(>F)    
## 1  18526     8253.7                                
## 2  18525     7741.2  1 512.48  1226.4 < 2.2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##   model      AIC
## 1     1 68451.26
## 2     2 67265.37
## 
## Formula: G_MassBal_MgHaYr ~ (1 + (MEASTIME_avg - 1990) * tau/100) * (1 - 
##     alpha * B_L_prop) * (a + b * exp(-((log(STDAGE_t1/c))/d)^2))
## 
## Parameters:
##       Estimate Std. Error t value Pr(>|t|)    
## tau    1.39928    0.17845   7.841 4.70e-15 ***
## alpha  0.82379    0.02151  38.294  < 2e-16 ***
## a      1.10879    0.26512   4.182 2.90e-05 ***
## b      1.24528    0.24934   4.994 5.96e-07 ***
## c     22.86693    0.96168  23.778  < 2e-16 ***
## d      1.78699    0.29232   6.113 9.97e-10 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.6464 on 18525 degrees of freedom
## 
## Algorithm "port", convergence message: relative convergence (4)
##   (4154 observations deleted due to missingness)

summary

  • simple tau model: fits
  • alpha model: fits

predict and plot

## Warning: Removed 1843 rows containing missing values (`geom_point()`).

plotting 2

221 - Eastern Broadleaf Forest

model selection 1

## Analysis of Variance Table
## 
## Model 1: G_MassBal_MgHaYr ~ (1 + (MEASTIME_avg - 1990) * tau/100) * (a + b * exp(-((log(STDAGE_t1/c))/d)^2))
## Model 2: G_MassBal_MgHaYr ~ (1 + (MEASTIME_avg - 1990) * tau/100) * (1 - alpha * B_L_prop) * (a + b * exp(-((log(STDAGE_t1/c))/d)^2))
##   Res.Df Res.Sum Sq Df Sum Sq F value    Pr(>F)    
## 1   6924     2689.6                                
## 2   6923     2569.8  1 119.77  322.64 < 2.2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##   model      AIC
## 1     1 30842.27
## 2     2 30528.64
## 
## Formula: G_MassBal_MgHaYr ~ (1 + (MEASTIME_avg - 1990) * tau/100) * (1 - 
##     alpha * B_L_prop) * (a + b * exp(-((log(STDAGE_t1/c))/d)^2))
## 
## Parameters:
##       Estimate Std. Error t value Pr(>|t|)    
## tau   -0.44591    0.15003  -2.972  0.00297 ** 
## alpha  0.76139    0.03963  19.212  < 2e-16 ***
## a      0.00000   30.61728   0.000  1.00000    
## b      4.40702   30.60200   0.144  0.88550    
## c     38.99102    8.01905   4.862 1.19e-06 ***
## d      2.81200   10.68362   0.263  0.79240    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.6093 on 6923 degrees of freedom
## 
## Algorithm "port", convergence message: relative convergence (4)
##   (378 observations deleted due to missingness)

summary

  • simple tau model: fits
  • alpha model: fits

predict and plot

## Warning: Removed 25 rows containing missing values (`geom_point()`).

plotting 2

222 - Midwest Broadleaf Forest

model selection 1

## Analysis of Variance Table
## 
## Model 1: G_MassBal_MgHaYr ~ (1 + (MEASTIME_avg - 1990) * tau/100) * (a + b * exp(-((log(STDAGE_t1/c))/d)^2))
## Model 2: G_MassBal_MgHaYr ~ (1 + (MEASTIME_avg - 1990) * tau/100) * (1 - alpha * B_L_prop) * (a + b * exp(-((log(STDAGE_t1/c))/d)^2))
##   Res.Df Res.Sum Sq Df Sum Sq F value    Pr(>F)    
## 1   4757     1856.5                                
## 2   4756     1762.7  1 93.856  253.24 < 2.2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##   model      AIC
## 1     1 19612.03
## 2     2 19366.98
## 
## Formula: G_MassBal_MgHaYr ~ (1 + (MEASTIME_avg - 1990) * tau/100) * (1 - 
##     alpha * B_L_prop) * (a + b * exp(-((log(STDAGE_t1/c))/d)^2))
## 
## Parameters:
##       Estimate Std. Error t value Pr(>|t|)    
## tau    0.21663    0.23614   0.917    0.359    
## alpha  0.77245    0.04446  17.373  < 2e-16 ***
## a      2.57655    0.18696  13.781  < 2e-16 ***
## b      0.81266    0.14950   5.436 5.73e-08 ***
## c     52.88239    2.44781  21.604  < 2e-16 ***
## d      0.77801    0.14915   5.216 1.90e-07 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.6088 on 4756 degrees of freedom
## 
## Algorithm "port", convergence message: relative convergence (4)
##   (1084 observations deleted due to missingness)

summary

  • simple tau model: fits
  • alpha model: fits

predict and plot

## Warning: Removed 501 rows containing missing values (`geom_point()`).

plotting 2

223 - Central Interior Broadleaf Forest

model selection 1

## Analysis of Variance Table
## 
## Model 1: G_MassBal_MgHaYr ~ (1 + (MEASTIME_avg - 1990) * tau/100) * (a + b * exp(-((log(STDAGE_t1/c))/d)^2))
## Model 2: G_MassBal_MgHaYr ~ (1 + (MEASTIME_avg - 1990) * tau/100) * (1 - alpha * B_L_prop) * (a + b * exp(-((log(STDAGE_t1/c))/d)^2))
##   Res.Df Res.Sum Sq Df Sum Sq F value    Pr(>F)    
## 1   8449     3574.8                                
## 2   8448     3491.5  1 83.323  201.61 < 2.2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##   model      AIC
## 1     1 35000.76
## 2     2 34803.38
## 
## Formula: G_MassBal_MgHaYr ~ (1 + (MEASTIME_avg - 1990) * tau/100) * (1 - 
##     alpha * B_L_prop) * (a + b * exp(-((log(STDAGE_t1/c))/d)^2))
## 
## Parameters:
##       Estimate Std. Error t value Pr(>|t|)    
## tau   -0.19097    0.14827  -1.288   0.1978    
## alpha  0.62818    0.04177  15.037  < 2e-16 ***
## a      1.68617    1.42899   1.180   0.2380    
## b      1.97015    1.41691   1.390   0.1644    
## c     28.57096    4.30586   6.635 3.44e-11 ***
## d      1.65615    0.92012   1.800   0.0719 .  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.6429 on 8448 degrees of freedom
## 
## Algorithm "port", convergence message: relative convergence (4)
##   (1552 observations deleted due to missingness)

summary

  • simple tau model: fits
  • alpha model: fits

predict and plot

## Warning: Removed 616 rows containing missing values (`geom_point()`).

plotting 2

231 - Southeastern Mixed Forest

model selection 1

## Analysis of Variance Table
## 
## Model 1: G_MassBal_MgHaYr ~ (1 + (MEASTIME_avg - 1990) * tau/100) * (a + b * exp(-((log(STDAGE_t1/c))/d)^2))
## Model 2: G_MassBal_MgHaYr ~ (1 + (MEASTIME_avg - 1990) * tau/100) * (1 - alpha * B_L_prop) * (a + b * exp(-((log(STDAGE_t1/c))/d)^2))
##   Res.Df Res.Sum Sq Df Sum Sq F value    Pr(>F)    
## 1  12080     5704.0                                
## 2  12079     4963.9  1 740.08  1800.9 < 2.2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##   model      AIC
## 1     1 60303.17
## 2     2 58625.69
## 
## Formula: G_MassBal_MgHaYr ~ (1 + (MEASTIME_avg - 1990) * tau/100) * (1 - 
##     alpha * B_L_prop) * (a + b * exp(-((log(STDAGE_t1/c))/d)^2))
## 
## Parameters:
##       Estimate Std. Error t value Pr(>|t|)    
## tau    1.54440    0.18218   8.477   <2e-16 ***
## alpha  0.90652    0.01895  47.844   <2e-16 ***
## a      2.98244    0.12403  24.045   <2e-16 ***
## b      1.91633    0.09835  19.485   <2e-16 ***
## c     17.48671    0.48145  36.321   <2e-16 ***
## d      0.98628    0.06672  14.783   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.6411 on 12079 degrees of freedom
## 
## Algorithm "port", convergence message: relative convergence (4)
##   (759 observations deleted due to missingness)

summary

  • simple tau model: fits
  • alpha model: fits

predict and plot

## Warning: Removed 96 rows containing missing values (`geom_point()`).

plotting 2

232 - Outer Coastal Plain Mixed Forest

model selection 1

## Analysis of Variance Table
## 
## Model 1: G_MassBal_MgHaYr ~ (1 + (MEASTIME_avg - 1990) * tau/100) * (a + b * exp(-((log(STDAGE_t1/c))/d)^2))
## Model 2: G_MassBal_MgHaYr ~ (1 + (MEASTIME_avg - 1990) * tau/100) * (1 - alpha * B_L_prop) * (a + b * exp(-((log(STDAGE_t1/c))/d)^2))
##   Res.Df Res.Sum Sq Df Sum Sq F value    Pr(>F)    
## 1  12420     8127.3                                
## 2  12419     7172.3  1 954.97  1653.5 < 2.2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##   model     AIC
## 1     1 62939.2
## 2     2 61388.1
## 
## Formula: G_MassBal_MgHaYr ~ (1 + (MEASTIME_avg - 1990) * tau/100) * (1 - 
##     alpha * B_L_prop) * (a + b * exp(-((log(STDAGE_t1/c))/d)^2))
## 
## Parameters:
##       Estimate Std. Error t value Pr(>|t|)    
## tau    1.65220    0.21855    7.56 4.32e-14 ***
## alpha  0.89704    0.01912   46.91  < 2e-16 ***
## a      2.76534    0.10479   26.39  < 2e-16 ***
## b      1.87484    0.09535   19.66  < 2e-16 ***
## c     16.11438    0.43598   36.96  < 2e-16 ***
## d      0.81297    0.04623   17.58  < 2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.76 on 12419 degrees of freedom
## 
## Algorithm "port", convergence message: relative convergence (4)
##   (742 observations deleted due to missingness)

summary

  • simple tau model: fits
  • alpha model: fits

predict and plot

## Warning: Removed 129 rows containing missing values (`geom_point()`).

plotting 2

234 - Lower Mississippi Riverine Forest

model selection 1

## Analysis of Variance Table
## 
## Model 1: G_MassBal_MgHaYr ~ (1 + (MEASTIME_avg - 1990) * tau/100) * (a + b * exp(-((log(STDAGE_t1/c))/d)^2))
## Model 2: G_MassBal_MgHaYr ~ (1 + (MEASTIME_avg - 1990) * tau/100) * (1 - alpha * B_L_prop) * (a + b * exp(-((log(STDAGE_t1/c))/d)^2))
##   Res.Df Res.Sum Sq Df Sum Sq F value    Pr(>F)    
## 1   1271     759.14                                
## 2   1270     714.87  1 44.271  78.649 < 2.2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##   model      AIC
## 1     1 6448.107
## 2     2 6373.437
## 
## Formula: G_MassBal_MgHaYr ~ (1 + (MEASTIME_avg - 1990) * tau/100) * (1 - 
##     alpha * B_L_prop) * (a + b * exp(-((log(STDAGE_t1/c))/d)^2))
## 
## Parameters:
##       Estimate Std. Error t value Pr(>|t|)    
## tau    0.85607    0.72485   1.181 0.237812    
## alpha  0.78091    0.07943   9.831  < 2e-16 ***
## a      3.41726    0.47753   7.156  1.4e-12 ***
## b      1.74467    0.45124   3.866 0.000116 ***
## c     18.45899    2.24438   8.225  4.8e-16 ***
## d      0.69423    0.18461   3.761 0.000177 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.7503 on 1270 degrees of freedom
## 
## Algorithm "port", convergence message: relative convergence (4)
##   (68 observations deleted due to missingness)

summary

  • simple tau model: fits
  • alpha model: fits

predict and plot

## Warning: Removed 17 rows containing missing values (`geom_point()`).

plotting 2

242 - Pacific Lowland Mixed Forest

model selection 1

summary

  • simple tau model: does not fit
  • alpha model: does not fit

predict and plot

## [1] "cannot plot data with prediction"

plotting 2

251 - Prairie Parkland (Temperate)

model selection 1

## Analysis of Variance Table
## 
## Model 1: G_MassBal_MgHaYr ~ (1 + (MEASTIME_avg - 1990) * tau/100) * (a + b * exp(-((log(STDAGE_t1/c))/d)^2))
## Model 2: G_MassBal_MgHaYr ~ (1 + (MEASTIME_avg - 1990) * tau/100) * (1 - alpha * B_L_prop) * (a + b * exp(-((log(STDAGE_t1/c))/d)^2))
##   Res.Df Res.Sum Sq Df Sum Sq F value Pr(>F)    
## 1   1737     622.69                             
## 2   1736     616.67  1  6.024  16.958  4e-05 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##   model      AIC
## 1     1 6711.617
## 2     2 6696.682
## 
## Formula: G_MassBal_MgHaYr ~ (1 + (MEASTIME_avg - 1990) * tau/100) * (1 - 
##     alpha * B_L_prop) * (a + b * exp(-((log(STDAGE_t1/c))/d)^2))
## 
## Parameters:
##       Estimate Std. Error t value Pr(>|t|)    
## tau     0.6255     0.4418   1.416   0.1570    
## alpha   0.4154     0.0965   4.304 1.77e-05 ***
## a       1.8585     1.0940   1.699   0.0895 .  
## b       0.8413     1.0662   0.789   0.4302    
## c      42.8805     4.5384   9.448  < 2e-16 ***
## d       1.1374     1.0631   1.070   0.2848    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.596 on 1736 degrees of freedom
## 
## Algorithm "port", convergence message: relative convergence (4)
##   (548 observations deleted due to missingness)

summary

  • simple tau model: fits
  • alpha model: fits

predict and plot

## Warning: Removed 246 rows containing missing values (`geom_point()`).

plotting 2

255 - Prairie Parkland (Subtropical)

model selection 1

## Analysis of Variance Table
## 
## Model 1: G_MassBal_MgHaYr ~ (1 + (MEASTIME_avg - 1990) * tau/100) * (a + b * exp(-((log(STDAGE_t1/c))/d)^2))
## Model 2: G_MassBal_MgHaYr ~ (1 + (MEASTIME_avg - 1990) * tau/100) * (1 - alpha * B_L_prop) * (a + b * exp(-((log(STDAGE_t1/c))/d)^2))
##   Res.Df Res.Sum Sq Df Sum Sq F value    Pr(>F)    
## 1    670     743.09                                
## 2    669     721.23  1 21.866  20.282 7.883e-06 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##   model      AIC
## 1     1 3193.464
## 2     2 3175.304
## 
## Formula: G_MassBal_MgHaYr ~ (1 + (MEASTIME_avg - 1990) * tau/100) * (1 - 
##     alpha * B_L_prop) * (a + b * exp(-((log(STDAGE_t1/c))/d)^2))
## 
## Parameters:
##       Estimate Std. Error t value Pr(>|t|)    
## tau     1.1668     1.3234   0.882 0.378260    
## alpha   0.7372     0.1466   5.028 6.37e-07 ***
## a       0.7819     0.6694   1.168 0.243203    
## b       2.4068     0.8155   2.951 0.003275 ** 
## c      18.3942     1.9567   9.401  < 2e-16 ***
## d       1.2802     0.3703   3.457 0.000581 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 1.038 on 669 degrees of freedom
## 
## Algorithm "port", convergence message: relative convergence (4)
##   (39 observations deleted due to missingness)

summary

  • simple tau model: fits
  • alpha model: fits

predict and plot

## Warning: Removed 28 rows containing missing values (`geom_point()`).

plotting 2

261 - California Coastal Chaparral Forest and Shrub

model selection 1

summary

  • simple tau model: does not fit
  • alpha model: does not fit

predict and plot

## [1] "cannot plot data with prediction"

plotting 2

262 - California Dry Steppe

model selection 1

summary

  • simple tau model: does not fit
  • alpha model: does not fit

predict and plot

## [1] "cannot plot data with prediction"

plotting 2

263 - California Coastal Steppe - Mixed Forest and Redwood Forest

model selection 1

summary

  • simple tau model: does not fit
  • alpha model: does not fit

predict and plot

## [1] "cannot plot data with prediction"

plotting 2

313 - Colorado Plateau Semi-Desert

model selection 1

summary

  • simple tau model: does not fit
  • alpha model: does not fit

predict and plot

## [1] "cannot plot data with prediction"

plotting 2

315 - Southwest Plateau and Plains Dry Steppe and Shrub

model selection 1

summary

  • simple tau model: does not fit
  • alpha model: does not fit

predict and plot

## [1] "cannot plot data with prediction"

plotting 2

321 - Chihuahuan Semi-Desert

model selection 1

summary

  • simple tau model: does not fit
  • alpha model: does not fit

predict and plot

## [1] "cannot plot data with prediction"

plotting 2

322 - American Semidesert and Desert

model selection 1

summary

  • simple tau model: does not fit
  • alpha model: does not fit

predict and plot

## [1] "cannot plot data with prediction"

plotting 2

331 - Great Plains/Palouse Dry Steppe

model selection 1

summary

  • simple tau model: does not fit
  • alpha model: does not fit

predict and plot

## [1] "cannot plot data with prediction"

plotting 2

332 - Great Plains Steppe

model selection 1

summary

  • simple tau model: does not fit
  • alpha model: does not fit

predict and plot

## [1] "cannot plot data with prediction"

plotting 2

341 - Intermountain Semi-desert & Desert

model selection 1

summary

  • simple tau model: does not fit
  • alpha model: does not fit

predict and plot

## [1] "cannot plot data with prediction"

plotting 2

342 - Intermountain Semi-Desert

model selection 1

summary

  • simple tau model: does not fit
  • alpha model: does not fit

predict and plot

## [1] "cannot plot data with prediction"

plotting 2

411 - Everglades

model selection 1

summary

  • simple tau model: does not fit
  • alpha model: does not fit

predict and plot

## [1] "cannot plot data with prediction"

plotting 2

M211 - Adirondack-New England Mixed forest - Coniferous Forest - Alpine Meadow

model selection 1

## Analysis of Variance Table
## 
## Model 1: G_MassBal_MgHaYr ~ (1 + (MEASTIME_avg - 1990) * tau/100) * (a + b * exp(-((log(STDAGE_t1/c))/d)^2))
## Model 2: G_MassBal_MgHaYr ~ (1 + (MEASTIME_avg - 1990) * tau/100) * (1 - alpha * B_L_prop) * (a + b * exp(-((log(STDAGE_t1/c))/d)^2))
##   Res.Df Res.Sum Sq Df Sum Sq F value    Pr(>F)    
## 1   6752     1843.1                                
## 2   6751     1736.0  1 107.05  416.31 < 2.2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##   model      AIC
## 1     1 25591.01
## 2     2 25188.68
## 
## Formula: G_MassBal_MgHaYr ~ (1 + (MEASTIME_avg - 1990) * tau/100) * (1 - 
##     alpha * B_L_prop) * (a + b * exp(-((log(STDAGE_t1/c))/d)^2))
## 
## Parameters:
##       Estimate Std. Error t value Pr(>|t|)    
## tau    0.91157    0.21485   4.243 2.24e-05 ***
## alpha  0.63278    0.02885  21.931  < 2e-16 ***
## a      1.85659    0.47443   3.913 9.19e-05 ***
## b      1.21316    0.45513   2.666 0.007705 ** 
## c     32.57864    2.02236  16.109  < 2e-16 ***
## d      1.58600    0.45499   3.486 0.000494 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.5071 on 6751 degrees of freedom
## 
## Algorithm "port", convergence message: relative convergence (4)
##   (21 observations deleted due to missingness)

summary

  • simple tau model: fits
  • alpha model: fits

predict and plot

## Warning: Removed 9 rows containing missing values (`geom_point()`).

plotting 2

M221 - Eastern Broadleaf Forest

model selection 1

## Analysis of Variance Table
## 
## Model 1: G_MassBal_MgHaYr ~ (1 + (MEASTIME_avg - 1990) * tau/100) * (a + b * exp(-((log(STDAGE_t1/c))/d)^2))
## Model 2: G_MassBal_MgHaYr ~ (1 + (MEASTIME_avg - 1990) * tau/100) * (1 - alpha * B_L_prop) * (a + b * exp(-((log(STDAGE_t1/c))/d)^2))
##   Res.Df Res.Sum Sq Df Sum Sq F value    Pr(>F)    
## 1   7755     3828.1                                
## 2   7754     3733.2  1 94.905  197.12 < 2.2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##   model      AIC
## 1     1 36463.27
## 2     2 36270.47
## 
## Formula: G_MassBal_MgHaYr ~ (1 + (MEASTIME_avg - 1990) * tau/100) * (1 - 
##     alpha * B_L_prop) * (a + b * exp(-((log(STDAGE_t1/c))/d)^2))
## 
## Parameters:
##       Estimate Std. Error t value Pr(>|t|)    
## tau    0.97063    0.23310   4.164 3.16e-05 ***
## alpha  0.82315    0.05549  14.835  < 2e-16 ***
## a      2.61737    0.22420  11.674  < 2e-16 ***
## b      1.26213    0.20978   6.016 1.86e-09 ***
## c     31.41373    4.34802   7.225 5.49e-13 ***
## d      0.92534    0.23823   3.884 0.000103 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.6939 on 7754 degrees of freedom
## 
## Algorithm "port", convergence message: relative convergence (4)
##   (426 observations deleted due to missingness)

summary

  • simple tau model: fits
  • alpha model: fits

predict and plot

## Warning: Removed 20 rows containing missing values (`geom_point()`).

plotting 2

M223 - Ozark Broadleaf Forest Meadow

model selection 1

## Analysis of Variance Table
## 
## Model 1: G_MassBal_MgHaYr ~ (1 + (MEASTIME_avg - 1990) * tau/100) * (a + b * exp(-((log(STDAGE_t1/c))/d)^2))
## Model 2: G_MassBal_MgHaYr ~ (1 + (MEASTIME_avg - 1990) * tau/100) * (1 - alpha * B_L_prop) * (a + b * exp(-((log(STDAGE_t1/c))/d)^2))
##   Res.Df Res.Sum Sq Df Sum Sq F value   Pr(>F)    
## 1    882     523.43                               
## 2    881     505.66  1 17.765  30.951 3.51e-08 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##   model      AIC
## 1     1 3696.088
## 2     2 3667.461
## 
## Formula: G_MassBal_MgHaYr ~ (1 + (MEASTIME_avg - 1990) * tau/100) * (1 - 
##     alpha * B_L_prop) * (a + b * exp(-((log(STDAGE_t1/c))/d)^2))
## 
## Parameters:
##       Estimate Std. Error t value Pr(>|t|)    
## tau     3.6254     1.8187   1.993 0.046522 *  
## alpha   0.9031     0.1494   6.045 2.20e-09 ***
## a       1.4214     0.3248   4.376 1.35e-05 ***
## b       0.9469     0.3297   2.872 0.004178 ** 
## c      32.2962     2.9490  10.951  < 2e-16 ***
## d       0.4015     0.1103   3.640 0.000289 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.7576 on 881 degrees of freedom
## 
## Algorithm "port", convergence message: relative convergence (4)
##   (6 observations deleted due to missingness)

summary

  • simple tau model: fits
  • alpha model: fits

predict and plot

## Warning: Removed 3 rows containing missing values (`geom_point()`).

plotting 2

M231 - Ouachita Mixed Forest

model selection 1

## Analysis of Variance Table
## 
## Model 1: G_MassBal_MgHaYr ~ (1 + (MEASTIME_avg - 1990) * tau/100) * (a + b * exp(-((log(STDAGE_t1/c))/d)^2))
## Model 2: G_MassBal_MgHaYr ~ (1 + (MEASTIME_avg - 1990) * tau/100) * (1 - alpha * B_L_prop) * (a + b * exp(-((log(STDAGE_t1/c))/d)^2))
##   Res.Df Res.Sum Sq Df Sum Sq F value    Pr(>F)    
## 1    952     508.31                                
## 2    951     472.57  1 35.732  71.906 < 2.2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##   model      AIC
## 1     1 4009.792
## 2     2 3942.037
## 
## Formula: G_MassBal_MgHaYr ~ (1 + (MEASTIME_avg - 1990) * tau/100) * (1 - 
##     alpha * B_L_prop) * (a + b * exp(-((log(STDAGE_t1/c))/d)^2))
## 
## Parameters:
##       Estimate Std. Error t value Pr(>|t|)    
## tau    4.49267    2.22922   2.015  0.04415 *  
## alpha  0.93086    0.09812   9.487  < 2e-16 ***
## a      1.41548    0.34868   4.060 5.32e-05 ***
## b      0.90555    0.30172   3.001  0.00276 ** 
## c     24.41902    1.76448  13.839  < 2e-16 ***
## d      0.33959    0.08496   3.997 6.91e-05 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.7049 on 951 degrees of freedom
## 
## Algorithm "port", convergence message: relative convergence (4)
##   (52 observations deleted due to missingness)

summary

  • simple tau model: fits
  • alpha model: fits

predict and plot

## Warning: Removed 4 rows containing missing values (`geom_point()`).

plotting 2

M242 - Cascade Mixed Forest

model selection 1

## Analysis of Variance Table
## 
## Model 1: G_MassBal_MgHaYr ~ (1 + (MEASTIME_avg - 1990) * tau/100) * (a + b * exp(-((log(STDAGE_t1/c))/d)^2))
## Model 2: G_MassBal_MgHaYr ~ (1 + (MEASTIME_avg - 1990) * tau/100) * (1 - alpha * B_L_prop) * (a + b * exp(-((log(STDAGE_t1/c))/d)^2))
##   Res.Df Res.Sum Sq Df Sum Sq F value    Pr(>F)    
## 1   3217     2988.6                                
## 2   3216     2870.6  1 118.03  132.23 < 2.2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##   model      AIC
## 1     1 17568.85
## 2     2 17441.02
## 
## Formula: G_MassBal_MgHaYr ~ (1 + (MEASTIME_avg - 1990) * tau/100) * (1 - 
##     alpha * B_L_prop) * (a + b * exp(-((log(STDAGE_t1/c))/d)^2))
## 
## Parameters:
##       Estimate Std. Error t value Pr(>|t|)    
## tau   -1.56033    0.30652  -5.090 3.78e-07 ***
## alpha  0.96775    0.07628  12.686  < 2e-16 ***
## a      5.95368    0.57338  10.384  < 2e-16 ***
## b      4.31980    0.84430   5.116 3.30e-07 ***
## c     35.17210    1.72271  20.417  < 2e-16 ***
## d      0.32134    0.05981   5.373 8.32e-08 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.9448 on 3216 degrees of freedom
## 
## Algorithm "port", convergence message: relative convergence (4)
##   (81 observations deleted due to missingness)

summary

  • simple tau model: fits
  • alpha model: fits

predict and plot

## Warning: Removed 38 rows containing missing values (`geom_point()`).

plotting 2

M261 - Sierran Steppe - Mixed Forest - Coniferous Forest - Alpine Meadow

model selection 1

## Analysis of Variance Table
## 
## Model 1: G_MassBal_MgHaYr ~ (1 + (MEASTIME_avg - 1990) * tau/100) * (a + b * exp(-((log(STDAGE_t1/c))/d)^2))
## Model 2: G_MassBal_MgHaYr ~ (1 + (MEASTIME_avg - 1990) * tau/100) * (1 - alpha * B_L_prop) * (a + b * exp(-((log(STDAGE_t1/c))/d)^2))
##   Res.Df Res.Sum Sq Df Sum Sq F value    Pr(>F)    
## 1   1691     1618.2                                
## 2   1690     1593.0  1 25.215  26.752 2.589e-07 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##   model      AIC
## 1     1 8793.291
## 2     2 8768.655
## 
## Formula: G_MassBal_MgHaYr ~ (1 + (MEASTIME_avg - 1990) * tau/100) * (1 - 
##     alpha * B_L_prop) * (a + b * exp(-((log(STDAGE_t1/c))/d)^2))
## 
## Parameters:
##       Estimate Std. Error t value Pr(>|t|)    
## tau    -2.4597     0.2437 -10.093  < 2e-16 ***
## alpha   0.6951     0.1251   5.558 3.16e-08 ***
## a       0.0000     4.5132   0.000   1.0000    
## b       8.0203     4.5315   1.770   0.0769 .  
## c      47.0468     7.6497   6.150 9.64e-10 ***
## d       2.6658     1.1090   2.404   0.0163 *  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.9709 on 1690 degrees of freedom
## 
## Algorithm "port", convergence message: relative convergence (4)
##   (297 observations deleted due to missingness)

summary

  • simple tau model: fits
  • alpha model: fits

predict and plot

## Warning: Removed 139 rows containing missing values (`geom_point()`).

plotting 2

M262 - Califormia Coastal Range = Coniferous Forest - Open woodland Shrub Meadow

model selection 1

summary

  • simple tau model: does not fit
  • alpha model: does not fit

predict and plot

## [1] "cannot plot data with prediction"

plotting 2

M313 - Arizona-New Mexico Mountains Semi-Desert - Open Woodland - Coniferous Forest - Alpine Meadow

model selection 1

## Analysis of Variance Table
## 
## Model 1: G_MassBal_MgHaYr ~ (1 + (MEASTIME_avg - 1990) * tau/100) * (a + b * exp(-((log(STDAGE_t1/c))/d)^2))
## Model 2: G_MassBal_MgHaYr ~ (1 + (MEASTIME_avg - 1990) * tau/100) * (1 - alpha * B_L_prop) * (a + b * exp(-((log(STDAGE_t1/c))/d)^2))
##   Res.Df Res.Sum Sq Df Sum Sq F value    Pr(>F)    
## 1    360     174.32                                
## 2    359     168.10  1 6.2251  13.295 0.0003055 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##   model      AIC
## 1     1 1014.707
## 2     2 1003.435
## 
## Formula: G_MassBal_MgHaYr ~ (1 + (MEASTIME_avg - 1990) * tau/100) * (1 - 
##     alpha * B_L_prop) * (a + b * exp(-((log(STDAGE_t1/c))/d)^2))
## 
## Parameters:
##       Estimate Std. Error t value Pr(>|t|)    
## tau    -2.4926     0.3002  -8.304 2.08e-15 ***
## alpha   0.5805     0.1477   3.931 0.000101 ***
## a       0.0000     5.0588   0.000 1.000000    
## b       3.3501     5.1051   0.656 0.512096    
## c      61.8680    17.6811   3.499 0.000526 ***
## d       2.0796     2.2905   0.908 0.364527    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.6843 on 359 degrees of freedom
## 
## Algorithm "port", convergence message: relative convergence (4)
##   (2 observations deleted due to missingness)

summary

  • simple tau model: fits
  • alpha model: fits

predict and plot

plotting 2

M331 - Southern Rocky Mountain Steppe - Open Woodland - Coniferous Forest - Alpine Meadow

model selection 1

## Analysis of Variance Table
## 
## Model 1: G_MassBal_MgHaYr ~ (1 + (MEASTIME_avg - 1990) * tau/100) * (a + b * exp(-((log(STDAGE_t1/c))/d)^2))
## Model 2: G_MassBal_MgHaYr ~ (1 + (MEASTIME_avg - 1990) * tau/100) * (1 - alpha * B_L_prop) * (a + b * exp(-((log(STDAGE_t1/c))/d)^2))
##   Res.Df Res.Sum Sq Df Sum Sq F value    Pr(>F)    
## 1   1723     1595.8                                
## 2   1722     1539.2  1 56.677   63.41 3.022e-15 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##   model      AIC
## 1     1 5203.597
## 2     2 5143.110
## 
## Formula: G_MassBal_MgHaYr ~ (1 + (MEASTIME_avg - 1990) * tau/100) * (1 - 
##     alpha * B_L_prop) * (a + b * exp(-((log(STDAGE_t1/c))/d)^2))
## 
## Parameters:
##       Estimate Std. Error t value Pr(>|t|)    
## tau   -0.78590    0.63025  -1.247 0.212583    
## alpha  0.60002    0.06600   9.091  < 2e-16 ***
## a      0.05448    0.69849   0.078 0.937841    
## b      1.87371    0.74000   2.532 0.011429 *  
## c     48.95939    3.63220  13.479  < 2e-16 ***
## d      1.99791    0.57426   3.479 0.000516 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.9454 on 1722 degrees of freedom
## 
## Algorithm "port", convergence message: relative convergence (4)
##   (29 observations deleted due to missingness)

summary

  • simple tau model: fits
  • alpha model: fits

predict and plot

## Warning: Removed 14 rows containing missing values (`geom_point()`).

plotting 2

M332 - Middle Rocky Mountain Steppe - Coniferous Forest - Alpine Meadow

model selection 1

## Analysis of Variance Table
## 
## Model 1: G_MassBal_MgHaYr ~ (1 + (MEASTIME_avg - 1990) * tau/100) * (a + b * exp(-((log(STDAGE_t1/c))/d)^2))
## Model 2: G_MassBal_MgHaYr ~ (1 + (MEASTIME_avg - 1990) * tau/100) * (1 - alpha * B_L_prop) * (a + b * exp(-((log(STDAGE_t1/c))/d)^2))
##   Res.Df Res.Sum Sq Df Sum Sq F value    Pr(>F)    
## 1   2510     2063.2                                
## 2   2509     1942.9  1 120.36  155.43 < 2.2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##   model      AIC
## 1     1 9308.899
## 2     2 9159.737
## Warning in `[<-.data.frame`(`*tmp*`, nls.param.df2$Code == "M332", , value =
## structure(list(: provided 26 variables to replace 25 variables
## 
## Formula: G_MassBal_MgHaYr ~ (1 + (MEASTIME_avg - 1990) * tau/100) * (1 - 
##     alpha * B_L_prop) * (a + b * exp(-((log(STDAGE_t1/c))/d)^2))
## 
## Parameters:
##       Estimate Std. Error t value Pr(>|t|)    
## tau   -0.47206    0.59179  -0.798    0.425    
## alpha  0.83144    0.05826  14.270  < 2e-16 ***
## a      0.00000    0.34184   0.000    1.000    
## b      2.48370    0.48494   5.122 3.26e-07 ***
## c     61.74217    4.48508  13.766  < 2e-16 ***
## d      2.29043    0.30910   7.410 1.72e-13 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.88 on 2509 degrees of freedom
## 
## Algorithm "port", convergence message: relative convergence (4)
##   (106 observations deleted due to missingness)

summary

  • simple tau model: fits
  • alpha model: fits

predict and plot

## Warning: Removed 55 rows containing missing values (`geom_point()`).

plotting 2

M333 - Northern Rocky Mountain Steppe - Coniferous Forest - Alpine Meadow

model selection 1

## Analysis of Variance Table
## 
## Model 1: G_MassBal_MgHaYr ~ (1 + (MEASTIME_avg - 1990) * tau/100) * (a + b * exp(-((log(STDAGE_t1/c))/d)^2))
## Model 2: G_MassBal_MgHaYr ~ (1 + (MEASTIME_avg - 1990) * tau/100) * (1 - alpha * B_L_prop) * (a + b * exp(-((log(STDAGE_t1/c))/d)^2))
##   Res.Df Res.Sum Sq Df Sum Sq F value    Pr(>F)    
## 1   1687     947.40                                
## 2   1686     857.14  1 90.266  177.55 < 2.2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##   model      AIC
## 1     1 6940.909
## 2     2 6773.494
## 
## Formula: G_MassBal_MgHaYr ~ (1 + (MEASTIME_avg - 1990) * tau/100) * (1 - 
##     alpha * B_L_prop) * (a + b * exp(-((log(STDAGE_t1/c))/d)^2))
## 
## Parameters:
##       Estimate Std. Error t value Pr(>|t|)    
## tau   -0.48912    0.66184  -0.739     0.46    
## alpha  0.87691    0.05808  15.099  < 2e-16 ***
## a      1.00833    0.22997   4.385 1.23e-05 ***
## b      3.25634    0.55305   5.888 4.71e-09 ***
## c     47.89490    1.79385  26.699  < 2e-16 ***
## d      1.35417    0.09350  14.484  < 2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.713 on 1686 degrees of freedom
## 
## Algorithm "port", convergence message: relative convergence (4)
##   (66 observations deleted due to missingness)

summary

  • simple tau model: fits
  • alpha model: fits

predict and plot

## Warning: Removed 34 rows containing missing values (`geom_point()`).

plotting 2

M334 - Black Hills Coniferous Forest

model selection 1

## Analysis of Variance Table
## 
## Model 1: G_MassBal_MgHaYr ~ (1 + (MEASTIME_avg - 1990) * tau/100) * (a + b * exp(-((log(STDAGE_t1/c))/d)^2))
## Model 2: G_MassBal_MgHaYr ~ (1 + (MEASTIME_avg - 1990) * tau/100) * (1 - alpha * B_L_prop) * (a + b * exp(-((log(STDAGE_t1/c))/d)^2))
##   Res.Df Res.Sum Sq Df  Sum Sq F value Pr(>F)
## 1    342     271.03                          
## 2    341     280.93  1 -9.9002 -12.017      1
##   model      AIC
## 1     1 1081.021
## 2     2 1095.471
## 
## Formula: G_MassBal_MgHaYr ~ (1 + (MEASTIME_avg - 1990) * tau/100) * (a + 
##     b * exp(-((log(STDAGE_t1/c))/d)^2))
## 
## Parameters:
##      Estimate Std. Error t value Pr(>|t|)    
## tau -1.347488   0.730018  -1.846  0.06578 .  
## a    1.631370   0.322270   5.062 6.79e-07 ***
## b   12.311276  10.354016   1.189  0.23525    
## c   56.399282   0.345131 163.414  < 2e-16 ***
## d    0.015805   0.005956   2.653  0.00834 ** 
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.8902 on 342 degrees of freedom
## 
## Algorithm "port", convergence message: relative convergence (4)
##   (104 observations deleted due to missingness)

summary

  • simple tau model: fits
  • alpha model: fits

predict and plot

## Warning: Using `size` aesthetic for lines was deprecated in ggplot2 3.4.0.
## ℹ Please use `linewidth` instead.
## Warning: Removed 48 rows containing missing values (`geom_point()`).
## Warning: Removed 1 rows containing missing values (`geom_segment()`).

plotting 2

## Warning: Removed 1 rows containing missing values (`geom_segment()`).

M341 - Nevada-Utah Mountains Semi-Desert - Coniferous Forest - Alpine Meadow

model selection 1

summary

  • simple tau model: does not fit
  • alpha model: does not fit

predict and plot

## [1] "cannot plot data with prediction"

plotting 2


Fitted parameters

Best / selected models by ecoprovince

Code Ecoregion Sel.Mod
211 Northeastern Mixed Forest 2
212 Laurentian Mixed Forest 2
221 Eastern Broadleaf Forest 2
222 Midwest Broadleaf Forest 2
223 Central Interior Broadleaf Forest 2
231 Southeastern Mixed Forest 2
232 Outer Coastal Plain Mixed Forest 2
234 Lower Mississippi Riverine Forest 2
242 Pacific Lowland Mixed Forest NA
251 Prairie Parkland (Temperate) 2
255 Prairie Parkland (Subtropical) 2
261 California Coastal Chaparral Forest and Shrub NA
262 California Dry Steppe NA
263 California Coastal Steppe - Mixed Forest and Redwood Forest NA
313 Colorado Plateau Semi-Desert NA
315 Southwest Plateau and Plains Dry Steppe and Shrub NA
321 Chihuahuan Semi-Desert NA
322 American Semidesert and Desert NA
331 Great Plains/Palouse Dry Steppe NA
332 Great Plains Steppe NA
341 Intermountain Semi-Desert and Desert NA
342 Intermountain Semi-Desert NA
411 Everglades NA
M211 Adirondack-New England Mixed forest - Coniferous Forest - Alpine Meadow 2
M221 Central Appalachian Broadleaf Forest - Coniferous Forest - Meadow 2
M223 Ozark Broadleaf Forest Meadow 2
M231 Ouachita Mixed Forest 2
M242 Cascade Mixed Forest 2
M261 Sierran Steppe - Mixed Forest - Coniferous Forest - Alpine Meadow 2
M262 California Coastal Range Coniferous Forest - Open Woodland - Shrub - Meadow NA
M313 Arizona-New Mexico Mountains Semi-Desert - Open Woodland - Coniferous Forest - Alpine Meadow 2
M331 Southern Rocky Mountain Steppe - Open Woodland - Coniferous Forest - Alpine Meadow 2
M332 Middle Rocky Mountain Steppe - Coniferous Forest - Alpine Meadow 2
M333 Northern Rocky Mountain Steppe - Coniferous Forest - Alpine Meadow 2
M334 Black Hills Coniferous Forest 1
M341 Nevada-Utah Mountains Semi-Desert - Coniferous Forest - Alpine Meadow NA

table by ecoprovince

Code Ecoregion region n.obs n.plots tau tau.variance tau.2.5 tau.97.5 alpha alpha.variance alpha.2.5 alpha.97.5 a a.2.5 a.97.5 b b.2.5 b.97.5 c c.2.5 c.97.5 d d.2.5 d.97.5
211 Northeastern Mixed Forest east 6884 2879 0.3122552 0.0309139 -0.0324135 0.6569239 0.6385218 0.0011229 0.5728311 0.7042125 0.0000000 -3.1378297 3.1378297 3.4328070 0.3083369 6.557277 34.47438 31.03604 37.91271 2.5431020 1.1065977 3.9796063
212 Laurentian Mixed Forest east 22685 9493 1.3992776 0.0318451 1.0494953 1.7490599 0.8237918 0.0004628 0.7816260 0.8659577 1.1087858 0.5891184 1.6284531 1.2452806 0.7565452 1.734016 22.86693 20.98194 24.75192 1.7869885 1.2140129 2.3599640
221 Eastern Broadleaf Forest east 7307 3560 -0.4459077 0.0225082 -0.7400075 -0.1518079 0.7613862 0.0015706 0.6836980 0.8390745 0.0000000 -60.0192561 60.0192561 4.4070188 -55.5822911 64.396329 38.99102 23.27122 54.71082 2.8120005 -18.1311648 23.7551659
222 Midwest Broadleaf Forest east 5846 2589 0.2166263 0.0557622 -0.2463179 0.6795706 0.7724515 0.0019769 0.6852858 0.8596172 2.5765512 2.2100221 2.9430802 0.8126565 0.5195614 1.105752 52.88239 48.08354 57.68123 0.7780129 0.4856186 1.0704073
223 Central Interior Broadleaf Forest east 10006 3860 -0.1909729 0.0219849 -0.4816243 0.0996784 0.6281816 0.0017451 0.5462934 0.7100697 1.6861654 -1.1150130 4.4873439 1.9701492 -0.8073439 4.747642 28.57096 20.13043 37.01150 1.6561479 -0.1475169 3.4598128
231 Southeastern Mixed Forest east 12844 5935 1.5444034 0.0331912 1.1872927 1.9015141 0.9065157 0.0003590 0.8693760 0.9436555 2.9824352 2.7393098 3.2255606 1.9163298 1.7235497 2.109110 17.48671 16.54300 18.43042 0.9862759 0.8554977 1.1170541
232 Outer Coastal Plain Mixed Forest east 13167 6463 1.6522029 0.0477639 1.2238119 2.0805939 0.8970390 0.0003657 0.8595565 0.9345215 2.7653363 2.5599380 2.9707346 1.8748420 1.6879393 2.061745 16.11438 15.25979 16.96897 0.8129738 0.7223488 0.9035987
234 Lower Mississippi Riverine Forest east 1344 759 0.8560691 0.5254062 -0.5659643 2.2781025 0.7809110 0.0063094 0.6250795 0.9367426 3.4172574 2.4804182 4.3540967 1.7446654 0.8594056 2.629925 18.45899 14.05589 22.86210 0.6942325 0.3320557 1.0564093
242 Pacific Lowland Mixed Forest west 85 85 NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA
251 Prairie Parkland (Temperate) east 2290 903 0.6254893 0.1951663 -0.2409806 1.4919592 0.4153713 0.0093131 0.2260938 0.6046489 1.8585086 -0.2871227 4.0041398 0.8413334 -1.2498538 2.932521 42.88047 33.97925 51.78170 1.1374160 -0.9476629 3.2224949
255 Prairie Parkland (Subtropical) east 714 318 1.1668247 1.7513681 -1.4316784 3.7653277 0.7371667 0.0214935 0.4493024 1.0250309 0.7818704 -0.5324699 2.0962108 2.4067541 0.8055229 4.007985 18.39423 14.55217 22.23628 1.2801691 0.5530109 2.0073273
261 California Coastal Chaparral Forest and Shrub west 26 26 NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA
262 California Dry Steppe west 0 0 NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA
263 California Coastal Steppe - Mixed Forest and Redwood Forest west 159 157 NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA
313 Colorado Plateau Semi-Desert west 218 218 NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA
315 Southwest Plateau and Plains Dry Steppe and Shrub west 4 4 NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA
321 Chihuahuan Semi-Desert west 9 9 NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA
322 American Semidesert and Desert west 3 3 NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA
331 Great Plains/Palouse Dry Steppe west 331 255 NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA
332 Great Plains Steppe west 232 128 NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA
341 Intermountain Semi-Desert and Desert west 66 64 NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA
342 Intermountain Semi-Desert west 124 123 NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA
411 Everglades east 96 63 NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA
M211 Adirondack-New England Mixed forest - Coniferous Forest - Alpine Meadow east 6778 3008 0.9115699 0.0461617 0.4903907 1.3327492 0.6327756 0.0008325 0.5762141 0.6893371 1.8565947 0.9265603 2.7866291 1.2131629 0.3209625 2.105363 32.57864 28.61417 36.54311 1.5860017 0.6940825 2.4779210
M221 Central Appalachian Broadleaf Forest - Coniferous Forest - Meadow east 8186 3765 0.9706339 0.0543361 0.5136927 1.4275750 0.8231535 0.0030790 0.7143811 0.9319259 2.6173655 2.1778674 3.0568635 1.2621315 0.8508994 1.673364 31.41373 22.89045 39.93702 0.9253407 0.4583535 1.3923278
M223 Ozark Broadleaf Forest Meadow east 893 348 3.6253706 3.3075234 0.0559592 7.1947820 0.9031479 NA 0.6099409 1.1963549 1.4213702 0.7839099 2.0588306 0.9469136 0.2997905 1.594037 32.29616 26.50819 38.08413 0.4015031 0.1850123 0.6179939
M231 Ouachita Mixed Forest east 1009 496 4.4926690 4.9694290 0.1179072 8.8674308 0.9308607 0.0096279 0.7383007 1.1234207 1.4154833 0.7312187 2.0997480 0.9055507 0.3134288 1.497673 24.41902 20.95630 27.88174 0.3395869 0.1728615 0.5063123
M242 Cascade Mixed Forest west 3303 3286 -1.5603333 0.0939549 -2.1613288 -0.9593377 0.9677509 0.0058191 0.8181824 1.1173195 5.9536792 4.8294557 7.0779027 4.3198009 2.6643712 5.975231 35.17210 31.79437 38.54982 0.3213441 0.2040698 0.4386184
M261 Sierran Steppe - Mixed Forest - Coniferous Forest - Alpine Meadow west 1993 1828 -2.4597236 0.0593881 -2.9377029 -1.9817442 0.6951360 0.0156415 0.4498356 0.9404364 0.0000000 -8.8521409 8.8521409 8.0202996 -0.8676141 16.908213 47.04680 32.04300 62.05061 2.6657677 0.4905137 4.8410218
M262 California Coastal Range Coniferous Forest - Open Woodland - Shrub - Meadow west 30 26 NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA
M313 Arizona-New Mexico Mountains Semi-Desert - Open Woodland - Coniferous Forest - Alpine Meadow west 367 367 -2.4926099 0.0901117 -3.0829541 -1.9022657 0.5805357 0.0218076 0.2901208 0.8709505 0.0000000 -9.9485137 9.9485137 3.3501094 -6.6895011 13.389720 61.86805 27.09645 96.63964 2.0796416 -2.4249326 6.5842158
M331 Southern Rocky Mountain Steppe - Open Woodland - Coniferous Forest - Alpine Meadow west 1757 1757 -0.7858964 0.3972178 -2.0220368 0.4502440 0.6000245 0.0043565 0.4705689 0.7294800 0.0544785 -1.3154965 1.4244535 1.8737103 0.4223100 3.325111 48.95939 41.83541 56.08337 1.9979081 0.8715839 3.1242324
M332 Middle Rocky Mountain Steppe - Coniferous Forest - Alpine Meadow west 2621 2611 -0.4720572 0.3502167 -1.6325063 0.6883919 0.8314380 0.0033948 0.7171860 0.9456901 0.0000000 -0.6703134 0.6703134 2.4836960 1.5327642 3.434628 61.74217 52.94733 70.53700 2.2904290 1.6843033 2.8965547
M333 Northern Rocky Mountain Steppe - Coniferous Forest - Alpine Meadow west 1758 1747 -0.4891160 0.4380352 -1.7872350 0.8090029 0.8769138 0.0033732 0.7629994 0.9908282 1.0083320 0.5572708 1.4593932 3.2563428 2.1716069 4.341079 47.89489 44.37648 51.41331 1.3541714 1.1707928 1.5375501
M334 Black Hills Coniferous Forest west 451 179 -1.3474876 0.5329266 -2.7833784 0.0884032 NA NA NA NA 1.6313704 0.9974897 2.2652510 12.3112756 -8.0542931 32.676844 56.39928 55.72043 57.07813 0.0158050 0.0040891 0.0275209
M341 Nevada-Utah Mountains Semi-Desert - Coniferous Forest - Alpine Meadow west 220 220 NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA

parameter variance co-variance

plot tau

map

## OGR data source with driver: ESRI Shapefile 
## Source: "C:\Users\hogan.jaaron\Dropbox\FIA_R\Mapping\S_USA.EcoMapProvinces\S_USA.EcoMapProvinces.shp", layer: "S_USA.EcoMapProvinces"
## with 37 features
## It has 17 fields
## Integer64 fields read as strings:  PROVINCE_ PROVINCE_I
## Warning: package 'ggnewscale' was built under R version 4.2.1
## Warning: `aes_string()` was deprecated in ggplot2 3.0.0.
## ℹ Please use tidy evaluation ideoms with `aes()`
## Warning: The `size` argument of `element_line()` is deprecated as of ggplot2 3.4.0.
## ℹ Please use the `linewidth` argument instead.
## Warning in grid.Call(C_stringMetric, as.graphicsAnnot(x$label)): font family not
## found in Windows font database

## Warning in grid.Call(C_stringMetric, as.graphicsAnnot(x$label)): font family not
## found in Windows font database
## Warning in grid.Call(C_textBounds, as.graphicsAnnot(x$label), x$x, x$y, : font
## family not found in Windows font database

## Warning in grid.Call(C_textBounds, as.graphicsAnnot(x$label), x$x, x$y, : font
## family not found in Windows font database

## Warning in grid.Call(C_textBounds, as.graphicsAnnot(x$label), x$x, x$y, : font
## family not found in Windows font database

## Warning in grid.Call(C_textBounds, as.graphicsAnnot(x$label), x$x, x$y, : font
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## Warning in grid.Call(C_textBounds, as.graphicsAnnot(x$label), x$x, x$y, : font
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plot alpha (biomass growth compensation effect)

plot a coefficient

## Warning: Removed 15 rows containing missing values (`geom_point()`).

plot b coefficient

## Warning: Removed 15 rows containing missing values (`geom_point()`).

plot c coefficient

## Warning: Removed 1 rows containing missing values (`geom_hline()`).
## Warning: Removed 15 rows containing missing values (`geom_point()`).

plot d coefficient

## Warning: Removed 15 rows containing missing values (`geom_point()`).